Can nonlinear dynamics contribute to chatter suppression? G

1
Can nonlinear dynamics contribute to chatter
suppression?Gábor Stépán
Department of Applied MechanicsBudapest
University of Technology and Economics
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Contents

• Motivation high-speed milling
• Physical background Periodically constrained
  inverted pendulum, and the swing Delayed PD
  control of the inverted pendulum Unstable
  periodic motion in stick-slip
• Periodic delayed oscillators delayed Mathieu
  equ.
• Nonlinear vibrations of cutting processes
• State dependent regenerative effect

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Motivation Chatter

• (high frequency) machine tool vibration
• Chatter is the most obscure and delicate of
  all problems facing the machinist probably no
  rules or formulae can be devised which will
  accurately guide the machinist in taking maximum
  cuts and speeds possible without producing
  chatter.
• 
  (Taylor, 1907).
• (Moon, Johnson, 1996)

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Efficiency of cutting

• Specific amount of material cut within a certain
  time
• where
• w chip width
• h chip thickness
• O cutting speed

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Modelling regenerative effect

• Mechanical model
• t time period of revolution
• Mathematical model

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Milling

• Mechanical model
• - number of cutting edgesin contact varies
  periodically with periodequal to the delay

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Contents

• Motivation high-speed milling
• Physical background Periodically constrained
  inverted pendulum, and the swing Delayed PD
  control of the inverted pendulum Unstable
  periodic motion in stick-slip
• Periodic delayed oscillators delayed Mathieu
  equ.
• Nonlinear vibrations of cutting processes
• State dependent regenerative effect

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Stabilizing inverted pendula

• Stephenson (1908) periodically forced pendulum
• Mathematical background
• Mathieu equation (1868)
• x 0 can be stable inLjapunov sense for ? lt 0

.
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Contents

• Motivation high-speed milling
• Physical background Periodically constrained
  inverted pendulum, and the swing Delayed PD
  control of the inverted pendulum Unstable
  periodic motion in stick-slip
• Periodic delayed oscillators delayed Mathieu
  equ.
• Nonlinear vibrations of cutting processes
• State dependent regenerative effect

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Contents

• Motivation high-speed milling
• Physical background Periodically constrained
  inverted pendulum, and the swing Delayed PD
  control of the inverted pendulum Unstable
  periodic motion in stick-slip
• Periodic delayed oscillators delayed Mathieu
  equ.
• Nonlinear vibrations of cutting processes
• State dependent regenerative effect

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Balancing with reflex delay

• 
  ? instability

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Contents

• Motivation high-speed milling
• Physical background Periodically constrained
  inverted pendulum, and the swing Delayed PD
  control of the inverted pendulum Unstable
  periodic motion in stick-slip
• Periodic delayed oscillators delayed Mathieu
  equ.
• Nonlinear vibrations of cutting processes
• Outlook Act wait control, periodic flow control

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Stickslip unstable periodic motion

• Experiments with brakepad-like arrangements(R
  Horváth, Budapest / Auburn)

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Contents

• Motivation high-speed milling
• Physical background Periodically constrained
  inverted pendulum, and the swing Delayed PD
  control of the inverted pendulum Unstable
  periodic motion in stick-slip
• Periodic delayed oscillators delayed Mathieu equ
• Nonlinear vibrations of cutting processes
• State dependent regenerative effect

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The delayed Mathieu equation

• Analytically constructed stability chart for
  testing numerical methods and algorithms
• Time delay and time periodicity are equal
• Damped oscillator
• Mathieu equation (1868)
• Delayed oscillator (1941 shimmy)

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The damped oscillator

• 
  stable
• 
  Maxwell(1865)
• 
  Routh (1877)
• 
  Hurwitz (1895)
• 
  Lienard
  Chipard (1917)

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Stability chart Mathieu equation

• 
  Floquet (1883)
• 
  Hill (1886)
• 
  Rayleigh(1887)
• 
  van der Pol
• 
  Strutt (1928)
• 
  Sinha (1992)
• Strutt Ince (1956) diagram
  swing(2000BC)
• Stephensons inverted pendulum (1908)

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The damped Mathieu equation
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The delayed oscillator

• Hsu Bhatt (1966)
• Stepan, Retarded Dynamical Systems (1989)

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Delayed oscillator with damping

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The delayed Mathieu stability charts

• 
  b0
• 
  e1
  e0

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Stability chart of delayed Mathieu

• 
  
• 
  Insperger,
  Stepan (2002)

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Contents

• Motivation high-speed milling
• Physical background Periodically constrained
  inverted pendulum, and the swing Delayed PD
  control of the inverted pendulum Unstable
  periodic motion in stick-slip
• Periodic delayed oscillators delayed Mathieu equ
• Nonlinear vibrations of cutting processes
• State dependent regenerative effect

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Modelling regenerative effect

• Mechanical model
• t time period of revolution
• Mathematical model

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Cutting force

• ¾ rule for nonlinear
• cutting force
• Cutting coefficient

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Linear analysis stability

• Dimensionless time
• Dimensionless chip width
• Dimensionless cutting speed
• TobiasTlusty, Altintas, BudakGradisek,
  Kalveram, Insperger

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Stability and bifurcations of turning

• 
  Subcritical Hopf
  bifurcation
  unstable vibrations
  around stable cutting

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The unstable periodic motion

• Shi, Tobias
• (1984)
• impactexperiment

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Case study thread cutting

• 
  m 346 kg
• 
  k97 N/µm
• 
  fn84.1 Hz
• 
  ?0.025
• 
  gge3.175mm

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Stability of thread cutting theoryexp.

• 
  O344 f/p
• 
  Quasi-periodic
• 
  vibrations
• 
  f184.5 Hz
• 
  f290.8 Hz

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Machined surface

• D176 mm, t 0.175 s

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Self-interrupted cutting
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High-speed milling

• 
  Parametrically
  interrupted cutting
• Low
  number of edges
• Low
  immersion
• Highly
  interrupted

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Highly interrupted cutting

• Two dynamics
• free-flight
• cutting with regenerative effect like an
  impact

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Stability chart of H-S milling

• 
  Sense of the
• 
  period
• 
  doubling
• 
  (or flip)
• 
  bifurcation?
• Linear model (Davies, Burns, Pratt,
  2000)Simulation (Balachandran, 2000)

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Subcritical flip bifurcation
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Bifurcation diagram chaos
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The fly-over effect

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Both period-2s unstable at b)
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Milling

• Mechanical model
• - number of cutting edgesin contact varies
  periodically with periodequal to the delay

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Phase space reconstruction

• A secondary B stable cutting C
  period-2 osc. Hopf (tooth pass
  exc.) (no fly-over!!!)
• noisy trajectory
  from measurement
  noise-free reconstructed trajectory
  cutting contact(Gradisek,Kalveram)

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The stable period-2 motion
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Lobes lenses with ?0.02

• 
  (Szalai, Stepan, 2006)

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with
?0.0038

• 
  (Insperger,
• 
  Mann, Bayly,
• 
  Stepan, 2002)

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Phase space reconstruction at A

• Stable milling Unstable
  milling with (Gradisek et al.)
  stable period-2(?) or
  quasi-periodic(?) oscillation

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Bifurcation diagram

• (Szalai, Stepan, 2005)

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Stability of up- and down-milling

• Stabilization by time-periodic parameters!
• Insperger, Mann, S, Bayly (2002)

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Contents

• Motivation high-speed milling
• Physical background Periodically constrained
  inverted pendulum, and the swing Delayed PD
  control of the inverted pendulum Unstable
  periodic motion in stick-slip
• Periodic delayed oscillators delayed Mathieu
  equ.
• Nonlinear vibrations of cutting processes
• State dependent regenerative effect

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State dependent regenerative effect

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State dependent regenerative effect

• State dependent time delay ? (x)
• Without state dependence
• With state dependence, the chip thickness is
• , fz feed
  rate,

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2 DoF mathematical model

• Linearisation at stationary cutting (Insperger,
  2006)
• Realistic range of parameters
• Characteristic function

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Stability chart comparison

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Conclusion

• Periodic modulation of cutting coefficient may
  result improvements in the stability, e.g., for
  high-speed milling, but
• It may also cause loss of stability via period-2
  oscillations, leading to lenses ( lobes), too.
• Subcriticality results reduction in safe
  chatter-free parameter domain for turning,
  milling,
• There is no nonlinear theory for state-dependent
  regenerative effect.
• Thank you for your attention!